In the mining industry, once material of value, such as ore situated below the surface of the ground, has been discovered, there exists a need to extract that material from the ground.
In the past, one more traditional method has been to use a relatively large open cut mining technique, whereby a great volume of waste material is removed from the mine site in order for the miners to reach the material considered of value. For example, referring to FIG. 1, the mine 101 is shown with its valuable material 102 situated at a distance below the ground surface 103. In the past, most of the (waste) material 104 had to be removed so that the valuable material 102 could be exposed and extracted from the mine 101. In the past, this waste material was removed in a series of progressive layers 105, which are ever diminishing in area, until the valuable material 102 was exposed for extraction. This is not considered to be an efficient mining process, as a great deal of waste material must be removed, stored and returned at a later time to the mine site 101, in order to extract the valuable material 102. It is desirable to reduce the volume of waste material that must be removed prior to extracting the valuable material.
The open cut method exemplified in FIG. 1 is viewed as particularly inefficient where the valuable resource is located to one side of the pit 105 of a desirable mine site 101. For example, FIG. 2 illustrates such a situation. The valuable material 102 is located to one side of the pit 105. In such a situation, it is not considered efficient to remove the waste material 104 from region 206, that is where the waste material is not located relatively close to the valuable material 102, but it is considered desirable to remove the waste material 104 from region 207, that is where it is located nearer to the valuable material 102. This then rings other considerations to the fore. For example, it would be desirable to determine the boundary between regions 206 and 207, so that not too much undesirable waste material is removed (region 206), yet enough is removed to ensure safety factors are considered, such as cave-ins, etc. This then leads to a further consideration of the need to design a ‘pit’ 105 with a relatively optimal design having consideration for the location of the valuable material, relative to the waste material and other issues, such as safety factors.
This further consideration has led to an analysis of pit design, and a technique of removing waste material and valuable material called ‘pushbacks’. This technique is illustrated in FIG. 3. Basically, the pit 105 is designed to an extent that the waste material 104 to be removed is minimised, but still enabling extraction of the valuable material 102. The technique uses ‘blocks’ 308 which represent smaller volumes of material. The area proximate the valuable material is divided into a number of blocks 308. It is then a matter of determining which blocks need to be removed in order to enable access to the valuable material 102. This determination of blocks 308′, then gives rise to the design or extent of the pit 105.
FIG. 3 represents the mine as a two dimensional area, however, it should be appreciated that the mine is a three dimensional area. Thus the blocks 308 to be removed are determined in phases, and cones, which represent more accurately a three dimensional ‘volume’ which volume will ultimately form the pit 105.
Further consideration can be given to the prior art situation illustrated in FIG. 3. Consideration should be given to the scheduling of the removal of blocks. In effect, what is the best order of block removal, when other business aspects such as time/value and discounted cash flows are taken into account? There is a need to find a relatively optimal order of block removal which gives a relatively maximum value for a relatively minimum effort/time.
Attempts have been made in the past to find this ‘optimum’ block order by determining which block(s) 308 should be removed relative to a ‘violation free’ order. Turning to the illustration in FIG. 4, a pit 105 is shown with valuable material 102. For the purposes of discussion, if it was desirable to remove block 414, then there is considered to be a ‘violation’ if we determined a schedule of block removal which started by removing block 414 or blocks 414, 412 & 413 before blocks 409, 410 and 411 were removed. In other words, a violation free schedule would seek to remove other blocks 409, 410, 411, 412 and 413 before block 414. (It is important to note that the block number does not necessarily indicate a preferential order of block removal).
It can also be seen that this block scheduling can be extended to the entire pit 105 in order to remove the waste material 104 and the valuable material 102. With this violation free order schedule in mind, prior art attempts have been made. FIG. 5 illustrates one such attempt. Taking the blocks of FIG. 4, the blocks are numbered and sorted according to a ‘mineable block order’ having regard to practical mining techniques and other mine factors, such as safety etc and is illustrated by table 515. The blocks in table 515 are then sorted 516 with regard to Net Present Value (NPV) and is based on push back design via Life-of-mine NPV sequencing, taking into account obtaining the most value block from the ground at the earliest time. To illustrate the NPV sorting, and turning again to FIG. 4, there is a question as which of blocks 409, 410 or 411 should be removed first. All three blocks can be removed from the point of view of the ability to mine them, but it may, for example, be more economic to remove block 410, before block 409. Removing blocks 409, 410 or 411 does not lead to ‘violations’ thus consideration can be given to the order of block removal which is more economic.
The NPV sorting is conducted in a manner which does not lead to violations of the ‘violation free order’, and provides a table 517 listing an ‘executable block order’. In other words, this prior art technique leads to a listing of blocks, in an order which determines their removal having regard to the ability to mine them, and the economic return for doing so.
Furthermore, a number of prior art techniques are considered to take a relatively simple view of the problems confronted by the mine designer in a ‘real world’ mine situation. For example, the size, complexity, nature of blocks, grade, slope and other engineering constraints and time taken to undertake a mining operation is often not fully taken into account in prior art techniques, leading to computational problems or errors in the mine design. Such errors can have significant financial and safety implications for the mine operator.
With regard to size, for example, prior art techniques fail to adequately take account of the size of a ‘block’. Depending on the size of the overall project, a ‘block’ may be quite large, taking some weeks, months or even years to mine. If this is the case, many assumptions made in prior art techniques fail to give sufficient accuracy for the modern day business environment.
Given that many of the mine designs are mathematically and computational complex, according to prior art techniques, if the size of the blocks were reduced for greater accuracy, the result will be that either the optimisation techniques used will be time in feasible (that is they will take an inordinately long time to complete), or other assumptions will have to be made concerning aspects of the mine design such as mining rates, processing rates, etc which will result in a decrease the accuracy of the mine design solution.
Some examples of commercial software do use mixed integer programming engines, however, the method of aggregating blocks requires further improvement. For example, it is considered that product ‘ECSI Maximiser’ by ECS International Pty Ltd uses a form of integer optimisation in their pushback design, but the optimisation is local in time, and it's problem formulation is considered too large to optimise globally over the life of a mine. Also the product ‘MineMax’ by MineMAX Ptd Ltd may be used to find a rudimentary optimal block sequencing with a mixed integer programming engine, however it is considered that it's method of aggregation does not respect slopes as is required in many situations. ‘MineMax’ also optimises locally in time, and not globally. Thus, where there are a large number of variables, the user must resort to subdividing the pit into separate sections, and perform separate optimisations on each section, and thus the optimisation is not global over the entire pit. It is considered desirable to have an optimisation that is global in both space and time.
Dynamic Programming Approach
The Lerchs-Grossman graph-theoretic algorithm (H. Lerchs & I. Grossman, “Optimum Design of Open-Pit Mines”, Transactions CIM, 1965) has been proved to give a relatively exact solution to the ultimate pit problem for an open-cut mine in three dimensions. Lerchs and Grossman also presents a dynamic programming approach to the problem in two dimensions, which has since been extended to three dimensions. However, solution of the three-dimensional graph theoretic algorithm is computationally inefficient in practical cases.
Linear Programming Approach
There is a linear program (LP), as presented by Underwood and Tolwinski (R. Underwood & B. Tolwinski, “A mathematical programming viewpoint for solving the ultimate pit problem”, EJOR, 1998). The availability of CPLEX (by Ilog, www.ilog.com) as a powerful LP solver motivates investigation of the LP approach to the ultimate pit problem.
The ultimate pit problem can be modelled as an integer program (IP), where a value of 1 is assigned to blocks included in the ultimate pit, and a value of 0 is assigned otherwise. The IP formulation for the problem is then as follows.
                    Let        ⁢                                  ⁢                  xi          =                                                                      1                  ,                                                                              if                  ⁢                                                                          ⁢                  block                  ⁢                                                                          ⁢                  i                  ⁢                                                                          ⁢                  is                  ⁢                                                                          ⁢                  included                  ⁢                                                                          ⁢                  in                  ⁢                                                                          ⁢                  the                  ⁢                                                                          ⁢                  ultimate                  ⁢                                                                          ⁢                  pit                                                                                                      0                  ,                                                            otherwise                                                    ⁢                                  ⁢        Then        ⁢                                  ⁢                  max          ⁢                                    ∑              i                        ⁢                                          v                i                            ⁢                              x                i                                                    ⁢                                  ⁢                  s          .          t          .                                          ⁢                                                                                          x                    i                                    ≤                                      x                    j                                                                                                ∀                                      j                    ∈                                          P                      ⁡                                              (                        i                        )                                                                                                                                                                                      x                    i                                    ∈                                      {                                          0                      ,                      1                                        }                                                                                                ∀                  i                                                                                        equation        ⁢                                  ⁢        1            
where
vi is the value assigned to block i
xi is the decision variable that designates whether block i is included in the ultimate pit or not
P(i) is the set of predecessor blocks of block i.
One objective is to maximise the net value of the material removed from the pit. Consider that the only constraints are precedence constraints, which enforce the requirement of safe wall slopes in the mine. In fact, this IP formulation has the property of total unimodularity. That is, the solution of the LP relaxation of this formulation will be integral (i.e. a set of 0's and 1's). This is an extremely desirable property for an integer program. It allows the IP to be solved as an LP using the Simplex method. This leads to greatly increased solution efficiency in terms of both CPU time and memory requirements. The exact mathematical formulation of the linear programming approach to the ultimate pit problem is therefore
                              max          ⁢                                    ∑              i                        ⁢                                          v                i                            ⁢                              x                i                                                    ⁢                                  ⁢                  s          .          t          .                                          ⁢                                                                                          x                    i                                    ≤                                      x                    j                                                                                                ∀                                      j                    ∈                                          P                      ⁡                                              (                        i                        )                                                                                                                                                                  0                  ≤                                      x                    i                                    ≤                  1                                                                              ∀                  i                                                                                        equation        ⁢                                  ⁢        2            
This is the ideal approach to solve the problem, and is considered to give the optimal solution in every case. Unfortunately, implementation of this exact formulation in CPLEX fails to solve for mining projects of realistic size. Since the optimisation is carried out at the block level, and there is a constraint for every precedence arc for each block, a very large number of constraints are applied. For example, if a mine has 198,917 blocks, and after CPLEX performs pre-processing on the formulation, the resulting reduced LP still has 1,676,003 constraints. CPLEX attempts to solve this formulation using the dual simplex method, generally recognized as the most efficient method for solving linear programs of this size. However, in the case of the example mine, CPLEX was found to crash during the solution process due to the very large number of constraints. Inversion of a constraint matrix of this magnitude (as required for converting solutions obtained from the dual simplex method back into primal space) is considered to place too great a memory requirement on the system.
There still exists a need, however, to improve prior art techniques. Given that mining projects, on the whole, are relatively large scale operations, even small improvements in prior art techniques can represent millions of dollars in savings, and/or greater productivity and/or safety.
It is desirable to provide an improved mine design.
An object of the present invention is to provide an improved method of pit design, which takes into account slope constraints.
Another object of the present invention is to provide an improved method of determining a cluster.
A further object of the present invention is to determine which blocks of a mine pit provide a relative maximum net value of material, also having regard to practical limitations, such as slope constraints.
Yet another object of the present invention is to alleviate at least one disadvantage of the prior art.
Any discussion of documents, devices, acts or knowledge in this specification is included to explain the context of the invention. It should not be taken as an admission that any of the material forms a part of the prior art base or the common general knowledge in the relevant art in Australia or elsewhere on or before the priority date of the disclosure and claims herein.